Wigner Functions for the Canonical Pair Angle and Orbital Angular Momentum with Applications to Quantum Information

Abstract:The concept of Wigner functions on classical phase spaces of physical systems, with the aim to describe the corresponding quantum mechanical statistical properties of those systems, is well-established for planar phase spaces. It is shown how that concept can be generalized to cylindrical ones {(q,p) in S^1 x R} like angle theta and orbital angularmomentum p of a classical rotator. Crucial new ingredients are the replacement of the angle theta by the equivalent pair (cos q, sin q) and the interpolation of the discontinuous quantum mechanical angular momenta m = 0, +1, -1,... in terms of the continuous classical momentum p by means of Whittaker's "cardinal" function, well known from interpolation and signal processing theories. Otherwise many structural properties of planar and cylindrical Wigner functions are very similar. The new framework is applied to elementary concepts of quantum information: qubits, cat states, 2-qubits like entangled EPR/Bell states.The results may be useful for the description and analysis of quantum information experiments with orbital angular momenta of (Laguerre-Gauss) light beams or electron beams.